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A Course on Large Deviations with an Introduction to Gibbs Measures

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Firas Rassoul-Agha; Timo Seppäläinen

This is an introductory course on the methods of computing asymptotics
of probabilities of rare events: the theory of large deviations. The
book combines large deviation theory with basic statistical mechanics,
namely Gibbs measures with their variational characterization and the
phase transition of the Ising model, in a text intended for a one
semester or quarter course.

The book begins with a straightforward approach to the key ideas
and results of large deviation theory in the context of independent
identically distributed random variables. This includes Cramér's
theorem, relative entropy, Sanov's theorem, process level large
deviations, convex duality, and change of measure arguments.

Dependence is introduced through the interactions potentials of
equilibrium statistical mechanics. The phase transition of the Ising
model is proved in two different ways: first in the classical way with
the Peierls argument, Dobrushin's uniqueness condition, and
correlation inequalities and then a second time through the
percolation approach.

Beyond the large deviations of independent variables and Gibbs
measures, later parts of the book treat large deviations of Markov
chains, the Gärtner-Ellis theorem, and a large deviation theorem of
Baxter and Jain that is then applied to a nonstationary process and a
random walk in a dynamical random environment.

The book has been used with students from mathematics, statistics,
engineering, and the sciences and has been written for a broad
audience with advanced technical training. Appendixes review basic
material from analysis and probability theory and also prove some of
the technical results used in the text.

Readership

Graduate students interested in probability, the theory of large
deviations, and statistical mechanics.

Reviews & Endorsements

It possesses a great value as an
introduction for more and more people (students and experienced
researchers) to these beautiful and highly active theories, as it is
written in a very motivating and fresh style...I think the authors did a
very good job to provide a text that can be taken as a base for an
interesting and useful lecture without much preparation or as a quick
but thorough introduction to this subject.

This is an introductory course on the methods of computing asymptotics
of probabilities of rare events: the theory of large deviations. The
book combines large deviation theory with basic statistical mechanics,
namely Gibbs measures with their variational characterization and the
phase transition of the Ising model, in a text intended for a one
semester or quarter course.

The book begins with a straightforward approach to the key ideas
and results of large deviation theory in the context of independent
identically distributed random variables. This includes Cramér's
theorem, relative entropy, Sanov's theorem, process level large
deviations, convex duality, and change of measure arguments.

Dependence is introduced through the interactions potentials of
equilibrium statistical mechanics. The phase transition of the Ising
model is proved in two different ways: first in the classical way with
the Peierls argument, Dobrushin's uniqueness condition, and
correlation inequalities and then a second time through the
percolation approach.

Beyond the large deviations of independent variables and Gibbs
measures, later parts of the book treat large deviations of Markov
chains, the Gärtner-Ellis theorem, and a large deviation theorem of
Baxter and Jain that is then applied to a nonstationary process and a
random walk in a dynamical random environment.

The book has been used with students from mathematics, statistics,
engineering, and the sciences and has been written for a broad
audience with advanced technical training. Appendixes review basic
material from analysis and probability theory and also prove some of
the technical results used in the text.

Graduate students interested in probability, the theory of large
deviations, and statistical mechanics.

Reviews:

It possesses a great value as an
introduction for more and more people (students and experienced
researchers) to these beautiful and highly active theories, as it is
written in a very motivating and fresh style...I think the authors did a
very good job to provide a text that can be taken as a base for an
interesting and useful lecture without much preparation or as a quick
but thorough introduction to this subject.